Why is it interesting?

Layman

The physical reason why ghosts may show up, is the non-local nature of the gauge- fixing procedure. If we demand, for instance, $$∂µAµ = 0 , \tag{(3.1)}$$ then the transition from some other gauge choice to this one requires knowledge of the field values of a given configuration over all of space-time. Since gauge transformations do not affect physical information, the information transmitted over space-time in order to realize the gauge (3.1), is unphysical. This is the explanation of the emergence of ghosts. We can avoid ghosts, if the gauge fixing at any point x in space-time, is done in such a way that no knowledge of the field values in points other than the point x is needed.TOPOLOGICAL ASPECTS OF QUANTUM CHROMODYNAMICS by Gerard ’t Hooft

Student

"One of the most insightful treatment of ghosts in quantum field theory appears in lecture notes for the Basko Polje Summer School (1976) by Benny Lautrup entitled Of Ghoulies and Ghosties." http://scipp.ucsc.edu/~haber/ph218/

Researcher

The group of gauge transformations $G$ means the bundles automorphisms which preserve the Lagrangian. (Source)

The gauge group is simply one fibre of the bundle, i.e. for example, $SU(2)$.

We denote the space of all connections by $A$. Now, to get physically sensible results we must be careful with these different notions:

Integration should therefore be carried out on the quotient space $\mathcal{G}=A/G$. Now $A$ is a linear space but $\mathcal{G}$ is only a manifold and has to be treated with more respect. Thus for integration purposes a Jacobian term arises which, in perturbation theory, gives rise to the well-known Faddeev-Popov "ghost" particles. Nonperturbatively it seems reasonable that global topological features of $\mathcal{G}$ will be relevant.

Geometrical Aspects of Gauge Theories by M. F. Atiyah

Examples

Example1

Example2:

History